interface to internal representation of OMDc data More...

#include <DataStore.hpp>

Inheritance diagram for OMDc::DataStore:

Public Member Functions
	DataStore (const Eigen::Ref< const Eigen::MatrixXd > S, const Eigen::Ref< const Eigen::MatrixXd > U, Eigen::Index r)
	standard contructor from snapshots and inputs More...

void	updateSystemMatrices (const Eigen::Ref< const Eigen::MatrixXd > L)
	recomputes internal representations for given modes More...

const Eigen::MatrixXd	getInputMatrix () const
	input matrix of linear system from internal representation More...

const Eigen::VectorXd	getInitialState () const
	computes intial state from identified system More...

Public Member Functions inherited from OMD::DataStore
const RefBlock	X () const
	read-access to left snaps More...

const RefBlock	Y () const
	read-access to right snaps More...

const Block	LTX () const
	read-access to projected left snaps More...

const Block	LTY () const
	read-access to projected right snaps More...

	DataStore (const Eigen::Ref< const Eigen::MatrixXd > S, Eigen::Index r)
	standard contructor from snapshots More...

void	updateSystemMatrices (const Eigen::Ref< const Eigen::MatrixXd > L)
	recomputes internal representations for given modes More...

const Eigen::MatrixXd	getSystemMatrix () const
	system matrix of linear system M = (L'YX'L) inv(L'XX'L) More...

const double	getResidual () const
	squared Frobenius residual: \|\|Y\|\| - \|\|(L'YX'L) inv(L'XX'L) (L'XY'L)\|\| More...

const Eigen::MatrixXd &	getProjector () const
	projector onto orthog. space of X'L More...

const Eigen::MatrixXd &	getYstar () const
	part of Y'L that is orthogonal to X'L as R from QR More...

Public Attributes
const Eigen::Index	p
	number of inputs

Public Attributes inherited from OMD::DataStore
const Eigen::Index	m
	number of snapshots

const Eigen::Index	r
	rank of reduced space

const Eigen::Index	n
	number of spatial locations (n > m)

Additional Inherited Members
Public Types inherited from OMD::DataStore
typedef Eigen::Block< const Eigen::MatrixXd, Eigen::Dynamic, Eigen::Dynamic, true >	Block
	short-hand for matrix block expression

typedef Eigen::Block< const Eigen::Ref< const Eigen::MatrixXd >, Eigen::Dynamic, Eigen::Dynamic, true >	RefBlock
	short-hand for referenced matrix block

Protected Attributes inherited from OMD::DataStore
const Eigen::Ref< const Eigen::MatrixXd >	S
	reference to snapshots (no data is stored)

Eigen::MatrixXd	LTS
	product of current mode and snapshots [r x m]

const double	normY
	Frobenius norm of left matrices.

Eigen::MatrixXd	W
	projector such that I - Z pinv(Z) = W W' with Z = X'L

Eigen::MatrixXd	XTL_qr_rep
	QR representation of X'L as [r x r] (upper triangular)

Eigen::MatrixXd	YTL_qr_rep_top
	QR representation of Y'L as [r x r] (top part)

Eigen::MatrixXd	YTL_qr_rep_bottom
	QR representation of Y'L as [r x r] (upper triangular)

Detailed Description

interface to internal representation of OMDc data

This class expands the OMD::DataStore by

the input data held by reference
the orthogonal projector of the inputs
a method for computing the identified input matrix
a method for the initial state

It is intended to be used with OMD::LineSearchFunctor . The OMDc cost is simplified to

$\Vert Y - L M L^\top X - LPU \Vert_F^2 = \Vert (I - L L^\top)Y \Vert_F^2 + \Vert L^\top Y Z (I - Z^\top X^\top L (L^\top X Z Z^\top X^\top L)^{-1} L^\top X Z \Vert_F^2$

where the $Z\in\mathbb{R}^{(m-1)\times (m-p-1)}$ denotes matrix representation of the projector onto the orthogonal space of the inputs. This representation is identical to the OMD cost but for the last term which is augmented by . Hence, the internal data is updated by such that the OMD::DataStore methods can be applied. E.g. the system matrix computes

$M = ( L^\top Y Z ) \, (L^\top X Z )^\dagger$

using SVD for the pseudo-inverse. It is internally simplified to

$M^\top = R_X^\dagger \, R_Y$

by the QR decomposition explained in DataStore::updateSystemMatrices()

Note: The residual, the system matrix, the state projector, and the the derivatives for conjugate gradient method are identical to OMD .

Constructor & Destructor Documentation

◆ DataStore()

OMDc::DataStore::DataStore	(	const Eigen::Ref< const Eigen::MatrixXd >	S,
		const Eigen::Ref< const Eigen::MatrixXd >	U,
		Eigen::Index	r
	)

standard contructor from snapshots and inputs

Parameters

S	snapshots [n x m]
U	inputs [p x (m-1)]
r	dimension of reduced system

Member Function Documentation

◆ getInitialState()

const Eigen::VectorXd OMDc::DataStore::getInitialState ( ) const

computes intial state from identified system

The initial reduced state is computed as the solution to

$\min_{a_0} \sum_{k=0}^{m-1} \Vert L^\top s_k - a_k \Vert_2^2 \quad \mathrm{s.t.} \,\, a_{k+1} = M a_k + P u_k$

The initial state is found such that the identified model evolves as closely as possible to the data set. Observe that the OMDc problem states the residual differently: it defines a stepwise error.

Note: The optimization problem can be solved by substituting the identified dynamics into the cost function, and then stacking block-wise powers of the system and input matrix. Then, the resulting problem in the initial state may be solved by the pseudo-inverse of the (large) stacked matrices. However, this implementation does not use stacked matrices but operates with small memory foot print directly on the state vectors.

Returns: initial reduced state as Eigen::Vector

◆ getInputMatrix()

const Eigen::MatrixXd OMDc::DataStore::getInputMatrix ( ) const

input matrix of linear system from internal representation

The input matrix is computed by

$P = ( L^\top Y - M L^\top X ) \, U^\dagger$

using SVD for the pseudo-inverse

Returns: input matrix [r x p]

◆ updateSystemMatrices()

void OMDc::DataStore::updateSystemMatrices ( const Eigen::Ref< const Eigen::MatrixXd > L )

recomputes internal representations for given modes

The internal representation of the data reads

$\begin{bmatrix} Z^\top X^\top L & Z^\top Y^\top L \end{bmatrix} = Q \, \begin{bmatrix} R_X & R_Y \\ 0 & Y^\ast \end{bmatrix}$

where the dimensions of the the QR decomposition are identical to OMD . Hence, the internal data stores are overwritten to contain the projection onto the space normal to the inputs.

Note: The internal data available through OMD::DataStore::getProjector() and OMD::DataStore::getYstar() contains the projected data, while the dimensions are not changed

Parameters

L	projection matrix [n x r]

The documentation for this class was generated from the following file:

/Users/studien/Documents/REPO/ModelReduction/omdc/libs/inc/OMDc/DataStore.hpp

Public Member Functions

Public Attributes